Question: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $p \neq 0$. $z = \dfrac{-4p + 20}{p^2 - 4p - 21} \div \dfrac{p - 5}{p + 3} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $z = \dfrac{-4p + 20}{p^2 - 4p - 21} \times \dfrac{p + 3}{p - 5} $ First factor the quadratic. $z = \dfrac{-4p + 20}{(p + 3)(p - 7)} \times \dfrac{p + 3}{p - 5} $ Then factor out any other terms. $z = \dfrac{-4(p - 5)}{(p + 3)(p - 7)} \times \dfrac{p + 3}{p - 5} $ Then multiply the two numerators and multiply the two denominators. $z = \dfrac{ -4(p - 5) \times (p + 3) } { (p + 3)(p - 7) \times (p - 5) } $ $z = \dfrac{ -4(p - 5)(p + 3)}{ (p + 3)(p - 7)(p - 5)} $ Notice that $(p - 5)$ and $(p + 3)$ appear in both the numerator and denominator so we can cancel them. $z = \dfrac{ -4(p - 5)\cancel{(p + 3)}}{ \cancel{(p + 3)}(p - 7)(p - 5)} $ We are dividing by $p + 3$ , so $p + 3 \neq 0$ Therefore, $p \neq -3$ $z = \dfrac{ -4\cancel{(p - 5)}\cancel{(p + 3)}}{ \cancel{(p + 3)}(p - 7)\cancel{(p - 5)}} $ We are dividing by $p - 5$ , so $p - 5 \neq 0$ Therefore, $p \neq 5$ $z = \dfrac{-4}{p - 7} ; \space p \neq -3 ; \space p \neq 5 $